In a nutshell:

In , a hyperplane is a line of the form (note that this is the standard form of a line).

In , a hyperplane is our ordinary plane of the form (note that this is the standard form of a plane).

Now, hyperplanes (in general) are planes in . They are of the form .

Unfortunately, there is no real way to visualize these things in 4 or more dimensions!

Does this little bit make sense?